Abstract:

Differential Equations (DEs) are among the most widely used mathematical tools in different area
of sciences. Solving DEs, either analytically or numerically, has become a centre of interest for
many mathematicians and a large variety of methods are nowadays available to solve DEs numerically.
When solving a mathematical problem numerically, evaluating the error is of high importance in
practice. Most of the methods already available for solving DEs are implemented with a mechanism
to perform a local error control.
However, in the real realm, it is common to require the numerical solution to approximate the
exact solution with accuracy to a certain number of decimal places or significant figures. To satisfy
this condition, we require the global error to be bounded by a specifically determined tolerance.
In this case, a local error control is not longer efficient. On one hand, controlling the local error
only cannot ensure that the required accuracy will be achieved. On the other hand, the use of
such approach requires the user to do some preliminary studies on the problem, and have deep
understanding of the method. Thus, we need a mechanism to control the global error in order to
compute the numerical solution for a user-supplied accuracy requirement in automatic mode.
The global error estimate calculated in the course of such a control can also be applied to improve
the numerical solution obtained. It is straight forward since, if the error estimate is found with
sufficiently high accuracy, we can just add it to the numerical solution to get a better approximation
to the exact value.
Thus, accurate evaluation of the the global error is crucial for the purpose mentioned above.
Several techniques are already developed to compute the global error of the numerical solution.
The most common algorithms include the Richardson extrapolation, Zadunaisky’s technique, Solving
for the correction, and Using two different methods.
These methods use two integrations to evaluate the global error, and the provided error estimate is
valid if the global error admits an expansion in powers of the step size. Another approach, known
as solving the linearised discrete variational equation, can also be used. This last differs from the
others by the use of a truncated Taylor expansion of the defect of the method to estimate the global
error; and solving the problem and estimating the error is roughly the same as one step of the
underlying method.
In this research, we will investigate numerically and compare the efficiency of different techniques
for global error evaluation applied to multistep methods for solving ordinary differential equations
(ODEs) and differential algebraic equations (DAEs). We will first study the global error evaluation
techniques in multistep formulas for solving ODEs on uniform grids. In the case of nonuniform
grids, both multistep methods with variable coefficients and interpolation-type multistep methods
will be considered. Then, we will extend our study to multistep methods for solving DAEs.
Theoretical background will accompany numerical works. The accuracy and reliability of the
global error evaluation strategies will be discussed and compared for different types of multistep
methods for solving ODEs and DAEs. We will analyse the efficiency in terms of accuracy obtained
and CPU time spent. For that, a series of numerical experiments is conducted on a set of test
problems with known solutions.